Optimal universal orbit bound for lattice representations

Determine the best possible universal bound on the number of automorphism orbits required to represent every finite group by a finite lattice, and determine whether requiring the representing lattice to have a regular orbit changes that optimal bound.

Background

The paper proves that every finite group has a finite lattice representation with at most 50 automorphism orbits and that the lattice can be chosen to have a regular orbit. The authors note that this bound is probably not optimal and that their construction may allow some orbits to be omitted. They also establish that four orbits do not suffice in general, since the cyclic group of order three cannot be represented by a lattice with at most four orbits, even without a regular-orbit requirement.

The unresolved problem is to identify the smallest universal orbit bound and to determine whether imposing the existence of a regular orbit increases that minimum. This asks for a sharp refinement of the bounded-orbit theorem proved in the paper.